The Galois Complexity of Graph Drawing Michael J. Bannister William - - PowerPoint PPT Presentation

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The Galois Complexity of Graph Drawing Michael J. Bannister William - - PowerPoint PPT Presentation

Sep 08, 2023 34 likes •570 views

The Galois Complexity of Graph Drawing Michael J. Bannister William E. Devanny David Eppstein Michael Goodrich Overview Motivation Galois theory Models of computation Results Undrawable graphs! Overview Motivation Galois theory Models

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The Galois Complexity of Graph Drawing

Michael J. Bannister William E. Devanny David Eppstein Michael Goodrich

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Overview

Galois theory Models of computation Results Motivation Undrawable graphs!

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Overview

Galois theory Models of computation Results Motivation Undrawable graphs! For some definition of undrawable

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Motivation

c x max cT x Simplex vs Interior Point

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Motivation

c max cT x Simplex Methods x

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Motivation

c max cT x Simplex Methods x

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Motivation

c max cT x Simplex Methods x

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Motivation

c max cT x Interior Point Methods x

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Motivation

c max cT x Interior Point Methods x

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Motivation

c max cT x Interior Point Methods x

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Motivation

c max cT x Interior Point Methods x

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Motivation

c max cT x Interior Point Methods x

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Motivation

Symbolic vs. Numerical algorithms Symbolic - manipulate mathematical expressions to obtain an exact answer for a problem Numerical - iteratively walk towards the answer, improving an approximate answer with each step Simplex method Interior point method

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Force Directed Graph Drawing

(Fruchterman and Reingold) fa(d) = d2/k fr(d) = k2/d Neighbors: All pairs: When the total force at each vertex is zero, we are at F. and

  • R. equilbrium

Also disqualify unstable/degenerate solutions

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Force Directed Graph Drawing

(Fruchterman and Reingold) fa(d) = d2/k fr(d) = k2/d Neighbors: All pairs: When the total force at each vertex is zero, we are at F. and

  • R. equilbrium

Also disqualify unstable/degenerate solutions

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Force Directed Graph Drawing

(Fruchterman and Reingold) fa(d) = d2/k fr(d) = k2/d Neighbors: All pairs: When the total force at each vertex is zero, we are at F. and

  • R. equilbrium

Also disqualify unstable/degenerate solutions

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SLIDE 17

Force Directed Graph Drawing

(Fruchterman and Reingold) fa(d) = d2/k fr(d) = k2/d Neighbors: All pairs: When the total force at each vertex is zero, we are at F. and

  • R. equilbrium

Also disqualify unstable/degenerate solutions

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SLIDE 18

Force Directed Graph Drawing

(Fruchterman and Reingold) fa(d) = d2/k fr(d) = k2/d Neighbors: All pairs: When the total force at each vertex is zero, we are at F. and

  • R. equilbrium

Also disqualify unstable/degenerate solutions length k

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SLIDE 19

Force Directed Graph Drawing

(Fruchterman and Reingold) fa(d) = d2/k fr(d) = k2/d Neighbors: All pairs: When the total force at each vertex is zero, we are at F. and

  • R. equilbrium

Also disqualify unstable/degenerate solutions

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SLIDE 20

Force Directed Graph Drawing

(Fruchterman and Reingold) fa(d) = d2/k fr(d) = k2/d Neighbors: All pairs: When the total force at each vertex is zero, we are at F. and

  • R. equilbrium

Also disqualify unstable/degenerate solutions

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SLIDE 21

Force Directed Graph Drawing

(Fruchterman and Reingold) fa(d) = d2/k fr(d) = k2/d Neighbors: All pairs: When the total force at each vertex is zero, we are at F. and

  • R. equilbrium

Also disqualify unstable/degenerate solutions length k

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Motivation - Graph Drawing

Many problems only have numerical algorithms Why? Fruchterman-Reingold Kamada-Kawai Spectral methods Circle packings

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Motivation - Graph Drawing

Many problems only have numerical algorithms Why? Fruchterman-Reingold Kamada-Kawai Spectral methods Circle packings Galois theory!

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Solving polynomials

Quadratics ax2 + bx + c = 0 x = −b±

√ b2−4ac 2a

⇒ Cubics ax3 + bx2 + cx + d = 0 ⇒ Substitute x = t −

b 3a

t3 + pt + q = 0 Substitute t = w −

p 3w

w6 + qw3 − p3

27 = 0

Quadratic in w3

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Solving polynomials

Quartic ax4 + bx3 + cx2 + dx + e = 0 ⇒ Still has a symbolic solution Very messy Quintic ax5 + bx4 + cx3 + dx2 + ex + f = 0 ⇒ ?

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Galois - a short biography

Born in France in 1811 Mathematician First to use group as a technical term Worked on polynomial equations Political activist Was expelled for his political opinions Imprisoned for threatening the King’s life Is shot and killed in the duel in 1832 Showed there is no quintic formula the night before

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Galois Theory

Draws a connection between groups and roots of polynomials where the group encodes the expressibility of the roots If the Galois group for a polynomial contains S5 as a subgroup, then the roots cannot be written using radicals π = 3.14159 . . . φ = 1.618... = 1+

√ 5 2

Written using radicals?

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Models of Computation

Algebraic computation tree A model in which each node makes a decision or computes a value using standard arithmetic functions of previous values x = 5 − y x > y ? True False Accept Reject

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Models of Computation

Quadratic computation tree Radical computation tree Bounded degree root computation tree

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Models of Computation

Quadratic computation tree An algebraic computation tree with square roots and complex conjugation Radical computation tree Bounded degree root computation tree

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Models of Computation

Quadratic computation tree An algebraic computation tree with square roots and complex conjugation Radical computation tree An algebraic computation tree with kth roots and complex conjugation for any integer k Bounded degree root computation tree

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Models of Computation

Quadratic computation tree An algebraic computation tree with square roots and complex conjugation Radical computation tree An algebraic computation tree with kth roots and complex conjugation for any integer k Bounded degree root computation tree An algebraic computation tree with taking roots of bounded degree polynomials and complex conjugation

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Models of Computation

Quadratic computation tree An algebraic computation tree with square roots and complex conjugation Radical computation tree An algebraic computation tree with kth roots and complex conjugation for any integer k Bounded degree root computation tree An algebraic computation tree with taking roots of bounded degree polynomials and complex conjugation Compass and straightedge model

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Approach

Polynomial Galois Group Graph Drawing Layout Expressibility in a Symbolic Model F & R Layout

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Approach

Polynomial Galois Group Graph Drawing Layout Expressibility in a Symbolic Model F & R Layout p(x) = xn + . . .

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Approach

Polynomial Galois Group Graph Drawing Layout Expressibility in a Symbolic Model F & R Layout p(x) = xn + . . . S5

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Approach

Polynomial Galois Group Graph Drawing Layout Expressibility in a Symbolic Model F & R Layout p(x) = xn + . . . S5 Cannot draw in a Radical Computation Tree

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Approach

Polynomial Galois Group Graph Drawing Layout Expressibility in a Symbolic Model F & R Layout p(x) = xn + . . . S5 Cannot draw in a Radical Computation Tree Lots of variables System of polynomials

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Approach

Polynomial Galois Group Graph Drawing Layout Expressibility in a Symbolic Model F & R Layout p(x) = xn + . . . S5 Cannot draw in a Radical Computation Tree Lots of variables System of polynomials ⇒ p(x) may have high degree

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Approach

Polynomial Galois Group Graph Drawing Layout Expressibility in a Symbolic Model F & R Layout p(x) = xn + . . . S5 Cannot draw in a Radical Computation Tree Lots of variables System of polynomials ⇒ p(x) may have high degree

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Approach

Polynomial Galois Group Graph Drawing Layout Expressibility in a Symbolic Model F & R Layout p(x) = xn + . . . S5 Cannot draw in a Radical Computation Tree Lots of variables System of polynomials ⇒ p(x) may have high degree Exploit symmetry to reduce degree

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Undrawable Graphs

Bounded degree root computation trees Fruchterman-Reingold Kamada-Kawai Spectral graph drawings Circle packings

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Undrawable Graphs

Bounded degree root computation trees p-cycles Fruchterman-Reingold Kamada-Kawai Spectral graph drawings Circle packings

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Undrawable Graphs

Bounded degree root computation trees p-cycles p-bipyramid Fruchterman-Reingold Kamada-Kawai Spectral graph drawings Circle packings

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Undrawable Graphs

Radical computation trees Fruchterman-Reingold Kamada-Kawai Multidimensional scaling

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Undrawable Graphs

Radical computation trees Fruchterman-Reingold Kamada-Kawai Multidimensional scaling

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Undrawable Graphs

Radical computation trees Fruchterman-Reingold Kamada-Kawai Multidimensional scaling

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Undrawable Graphs

Radical computation trees Fruchterman-Reingold Kamada-Kawai Multidimensional scaling

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Spectral graph drawings

Undrawable Graphs

(Adjacency/Transition matrix) (Laplacian matrix) Radical computation trees

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Undrawable Graphs

Circle packings Radical computation trees

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Summary

Lots of graph drawing uses numerical algorithms Why no symbolic algorithms? Galois theory! Graph drawing coordinates cannot be computed using radicals

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Summary Open Questions

Lots of graph drawing uses numerical algorithms Why no symbolic algorithms? Galois theory! Graph drawing coordinates cannot be computed using radicals Other graph drawing problems with no symbolic algorithms? Problems with arbitrarily high Sn?

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Summary Open Questions

Lots of graph drawing uses numerical algorithms Why no symbolic algorithms? Galois theory! Graph drawing coordinates cannot be computed using radicals Other graph drawing problems with no symbolic algorithms? Problems with arbitrarily high Sn?

Thank you